CUPERTINO, Calif.—Apple Inc.’s new iPhone X ties the future of its flagship device to facial-recognition technology that could alter how people interact with their gadgets—if the company can get it to work right.

Facial recognition is the flashiest new feature in the 10th-anniversary iPhone Apple unveiled on Tuesday. Called Face ID, it will be the primary tool to unlock the nearly $1,000 iPhone X, which is scheduled to start shipping Nov. 3. A camera system with depth sensors projects 30,000 infrared dots across a user’s...

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